Luke and Owen have $\$100$ each. Their friend offered to invest their money, promising to return a sum $r$ times as great as what they invested. Luke was suspicious, so he invested $\$10$ only, but Owen invested his entire $\$100$. Fortunately, the friend did indeed return a sum $r$ times as great to each. They decided to make another investment. This time, Owen invested all of the money returned to him, and Luke invested the money returned to him and the remaining $\$90$. Again, they got a sum $r$ times as great as what they invested. In the end, Owen had $\$337.50$ more than Luke. Write an equation in terms of $r$ that models the situation.
Explanation: The strategy We know that in the end, Owen had $\$337.50$ more than Luke. If we let $O$ denote Owen's final amount and let $L$ denote Luke's final amount, we obtain the equation $O=L+337.50$. Now, let's express $O$ and $L$ in terms of $r$. Expressing Owen's final amount With the first investment, Owen invested $\$100$ and received an amount $r$ times what was invested, or $100r$ dollars. Owen then invested this amount, $100r$ dollars, and again received an amount $r$ times what was invested, or $100r\cdot r=100r^2$ dollars back. Expressing Luke's final amount Luke invested $\$10$ at first and received $10r$ dollars back. Then he invested this and the remaining $\$90$. So his total second investment was $10r+90$ dollars, and so he received $(10r+90) r=10r^2+90r$ dollars back. Putting things together We found that $O=100r^2$ and $L=10r^2+90r$. Since $O=L+337.50$, we can substitute and find an equation in terms of $r$ that models the situation. The answer is: $ 100r^2=10r^2+90r+337.50$